100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Space Time in String Theory 339The Bekenstein-Hawking relation suggests however that metrical conceptsoriginate in quantum mechanics. Thus, we quantize the pixel variables,by the unique SO(d − 2) invariant formula that gives a single pixela finite number of states:[S a (n), S b (n)] + = δ ab .In writing S a (n) we have anticipated the fact that the whole causal diamondmust have a finite number of states. We view the surface of its holographicscreen as broken up into pixels, labelled by a finite set of integers n. Eachpixel has area equal to the logarithm of the dimension of the minimal representationof the above Clifford algebra s . For 11 dimensional space-time,this dimension is 256. We see an immediate connection to 11D SUGRA.The SO(d − 2) content of this representation is exactly that of the supergravitonmultiplet. Thus, in particle physics language, what we are sayingis that specifying a holographic screen at each point is specifying the directionof the momentum of the massless particles, which can penetrate thispixel, as well as the possible spin states of those particles. The particle languagereally becomes justified only in the limit of infinite causal diamonds,in asymptotically flat space 50 . In that limit, a new quantum number, thelongitudinal momentum of the massless particle, also arises, in a mannerreminiscent of Matrix Theory 12 .Physically, the operators of individual pixels are independent quantumdegrees of freedom and so the operators associated with differentpixels should commute with each other. However, the full set of (anti)-commutation relations is invariant under a Z k 2 subgroup of the classical projectiveinvariance of the CP equation: S a (n) → (−1) Fn S a (n). This shouldbe treated as a gauge symmetry, and we can do a Klein transformation tonew variables that satisfy t[S a (n), S b (m)] + = δ ab δ mn .An elegant way to describe the pixelation of the holographic screenis to replace the algebra of functions on the screen by a finite dimensionalassociative algebra. If we want to implement space time rotation symmetriess Our equation is written for the case where the minimal spinor representation of SO(d−2)is real.t For asymptotically flat space it is convenient to choose another gauge for the Z 2 symmetry.In that limit the label n encodes multiparticle states, as well as the momentaof individual particles. It is convenient to choose pixel operators for different particlesto commute, in order to agree with the standard multiparticle quantum mechanicsconventions.